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We are interested in the stability of two-dimensional localized structures in  reaction-diffusion systems subjected to a delayed feedback. We show that the presence of  the delay can induce complex behavior of the localized structures, including, e.g.,  spontaneous motion or breathing of the localized objects. In the case of spontaneous  motion, the corresponding order parameter equation for the position of the localized  structure is derived. In addition, numerical simulations are carried out showing good  agreement with the analytical predictions

Destabilization of Localized Structures in Reaction-Diffusion Systems Induced by Delayed Feedback